Our swim club is a social enterprise and charged £70 for 3 x 30 min private lessons for 5 yr old DS. We were able to share the lesson with another child which halved the price. The lessons took place in a private pool.
There are often threads on here about parent's frustration at DC's lack of progress at swimming. I am sure there is a market.
If I was looking for you I'd try google or the swimming body if it has a directory.
Good news! I am able to more details regarding one to one lessons, mainly for adults as daytime hours, but children very welcome too who may not be in full time education. I am now part of the "DC LEISURE" team. I will initially be based at: ROTHERHAM and MALTBY. I may also be available at WATH and ASTON-CUM-AUGHTON. Lessons are booked in blocks of 5 - 1/2 hour sessions. the cost is £86.25. £17.25 EACH If you require more details, ring one of these sites for more information. Thanks for getting in touch. JOANNE.